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Understanding confirmed predictions in quantum gravity 

Nige Cook 


1 February 2013 

Feynman's relativistic path integral replaces the non-relativistic 1 st quantisation indeterminancy principle (required when 
using a classical Coulomb field in quantum mechanics) with a simple physical mechanism, multipath interference between 
small mechanical interactions. Each mechanical interaction is represented by a Feynman Moller scattering diagram (Fig. 1) 
for a gauge boson emitted by one charge to strike an effective interaction cross-section of the other charge, a cross-sec- 
tionl that is proportional to the square of the interaction strength or running coupling. Each additional pair of vertices in 
a Feynman diagram reduce its relative contribution to the path integral by a factor of the coupling, so for a force with a very 
small couplings like observable (low energy) quantum gravity, only the 2-vertex Feynman diagram has an appreciable contribution, 
allowing a very simple calculation to check the observable (low energy) quantum gravity interaction strength. Evidence is 
given that quantum gravity arises from a repulsive U(l) gauge symmetry which also causes the cosmological acceleration. 

Radioactive "beta decay" (weak inter- 
action) as first formulated by Wolfgang 
Pauli in 1930 (note that an antineutrino 
emission or output is equivalent to a 
neutrino as input): 

A + 1 protons nucleus 

A protons 

Fermi's 1934 theory of beta decay as 
equivalent to a neutron-neutrino point- 
scattering event (like electron scatter- 
ing), but with an isospin-change: 

In the 1967 electroweak theory: 



Antineutrino Neutrino scattering 


Electron Neutrino 


Massive W + 
boson exchange 


Electron . 

Low energy Moller (2 vertex) force-mediation Feynman diagram: 

mechanical scattering. More complex Feynman diagrams have trivial 

contributions, since each extra pair of vertices reduces its contribution 

to the path integral by a factor of the tiny gravity coupling, a 



Internal line (gauge boson) force 
propagator = il {cf - wr) for spin-0 




(by convention) 

We know both momentum and 
mass-energy are conserved in 
interactions, so external line 
momentum propagators 
(delta vertex functions) are 
symmetrical, and therefore 
automatically cancel out. 


Fig. 1: Weak isospin theory and the electromagnetic low-energy Moller scattering Feynman diagram, which by "crossing- 
symmetry" has the same cross-section as Bhabha's/> +e attraction. In more complex Feynman diagrams, each additional 
pair of vertices (due to a loop) reduces the contribution of the diagram to the path integral by a coupling factor OC, thus 
making their contributions to the path integral trivial if OC is very small (observable gravity has the smallest coupling OC). 
For this reason, only the simplest diagram need be considered for low energy quantum gravity, where the theory can be 
easily checked by predicting the observable strength of gravity. These diagrams depict simple 2-vertex Moller type scatter- 
ing diagrams, where forces are physically caused by the momentum exchanged by gauge boson being transferred between 
the charges, like bullets delivering forces. Using Feynman's rules for Feynman diagram contributions to the perturbative 
expansion of the path integral, we correctly predicted the cosmological acceleration of the universe (published in 1996, two 
years ahead of observational confirmation). Gravity was correctly predicted as the asymmetry in the total isotropic inward 
graviton exchange force fi-otal = ma > cause d by a particle's gravitational coupling cross-section at distance R from the 

9 9 

observer, which blocks a fraction of F tota i equal to the fraction of sky blocked, <7/(47lR ), so F y ^ T = ma o/(47tR ). 


f General relativity: I 
M 1 smooth curvature ^2 

Field quanta interactions 
are simply ignored 

I Gravitons \ 

Time V \ Gravitons 

Spin-2 graviton theory 
ignores other masses 


Receding masses' 

^^ ^T "*^ * 

f Spin-1 gravitons\£yj r W'Y 
push masses | j 

together AA 2 

Spin-1 graviton theory shows that exchange of gravitons 
causes both dark energy and gravitation effects 

Fig. 2: the predominant interaction is the single propagator 
Feynman diagram with two vertices, in which the cross-section 
is proportional to the square of the coupling. This scaling uses 
Feynman's diagram calculating rules: two-vertex interaction 

probability and cross-section are proportional to (coupling) . 
It empirically extrapolates the cross-section of a proton for 
scattering a graviton independently of any assumed graviton spin: 

"gravition-proton "" "neutrino-proton (VN' ^ Fermi > 

(10" 11 ) (10" 38 /10" 5 ) 2 

10" 77 mb or 10" 80 barns (1 barn 

10 m ). Gj^j is the gravity coupling, in the same ^proton 
units as the weak coupling, Gp„ rm j and using the H = c — 1 con- 
vention. This approximation (protons are not fundamental) is the black 
hole event horizon si%e, %(2Gm. 


/c 2 ) 2 = 1.93 Xl0" 79 barns. 

(From Los Alamos Science, Summer/Fall 1984, p. 23, Table 1.) 

Neutrino-Proton Scattering 

v + p ~* v + p 

a ~ lO 41 mb 


Fermi ^proton 

2 = 2 l/2„2^ 


/(8M 2 )~10" 


Graviton-Proton Scattering 


a ~ 10" 77 mb 

Newtonian "proton 

2 _ 10 -38 


1. Feynman's rules (Fig. 2) prove that the graviton scatter cross-section area is: Gg-tzvitv = °weak (^Newton/ ^Fermi ) 
~ K(2GM/ ' ir ) ~ 2 X 10" barns. (The nuclear cross-section unit is: 1 barn = 10 m .) The probability of hitting cross- 

section G (in area units) from distance R when firing radiation out isotropically is <j/(47TR ), which is equal to the fraction 
of the total isotropic radiation which is actually received by cross-section CT. (See .) 

2. Isotropic cosmological acceleration a (the dark energy produced cosmological acceleration, first observed in 1998) of the 
universe's mass m produces from our perspective an effective radial outward force by Newton's 2nd law F = dpi dt ~ ma~ 

[3 X 1CP kg] [7 X 10 U ms ] ~ 2 X lO 43 N, and Newton's 3rd law of motion (the rocket principle), in our frame of ref- 
erence as observers of that acceleration predicts an equal and opposite (inward directed) reaction force (see Figures 2, 4, 5 

and 6 in ). Hence g~ Mc /(amR ), predicting a ~ c /(Gm) = 7 X 10" U ms" , or A = 

r/{GtiT ), which was confirmed by observations of supernovae two years later in 1998, so G and A are not independent 
as assumed in general relativity, but are instead interdependent. The Lambda-CDM FRW metric based on general relativity 
ignores this dynamic mechanism where the dark energy causes gravity, so it falsely treats A and G as independent variables. But 
since momentum is conserved, a falling apple cannot gain momentum (accelerate) from a purely "geometric spacetime" 
without a backreaction upon the field (Newton's 3rd law). (Objections to LeSage-type gravity like heating or drag are inapplica- 
ble to all off hell field quanta, which in the Casimir force push metal plates together without any heating or drag, while the 

small cross-section of <T ~ 2 X 10" barns prevents the LeSage "overlap" problem; see ). 

Boson exchanged between charges to produce fundamental interactions have been detected, e.g. weak bosons (Fig. 1) have 
been detected as neutral currents and their masses were found at CERN in 1983. It's a fact that Feynman's rules work for 

all three Standard Model forces, so their use for the fourth force, gravity, in Gptxrixsr = ^weak (^N^ ^Fermi ) ^ s based 

on the general, empircally confirmed Feynman rules. It's a fact that an acceleration of mass is a force by Newton's 2nd law (thus 

the outward acceleration of 3 X 10 kg in the universe constitutes a radial outward force ~ 2 X 10 N), and that Newton's 

3rd law predicts an equal inward force. It's not speculative to suggest that this inward radial force of 2 X 10 N is carried by 
gravitons, because this prediction was confirmed in 1998 by Saul Perlmutter's discovery of supernovae accelerating away 

from us with acceleration around 7x10 ms . This gravity prediction is not a speculation but a confirmed fact, based on 
consistent empirical physics. This predicted the cosmological acceleration in 1996, two years before confirmation by 
Perlmutter's automated software analysis of CCD telescope output, unlike spin-2 graviton speculation in unpredictive string theory. 

Implications for physics 

The application of general relativity to cosmology, i.e. the Friedmann-Robertson-Walker metric of general relativity, is plain 
wrong in the sense that it lacks the quantum gravity dynamics for gravity being the product of the surrounding accelera- 
tion of matter, so Friedmann cannot be used as a basis for analyzing the Hubble recession, big bang or any other large- 
scale (cosmological) features. General relativity is not an empirically validated prediction system for cosmology anyway 
since relies in any case upon ad hoc adjustable parameters, such as its cosmological constant which is not fixed by gneral 
relativity (unlike the situation in quantum gravity), but is adjusted to fit observations. However, general relativity does 
include energy conservation and corrects defects in Newtonian gravitation, allowing confirmed local (non-cosmological) 
predictions like the correct deflection of light by gravity, gravitational time dilation and redshift, etc. 

Production of the useful predictions from general relativity 

The basic method used by Einstein to arrive at the field equation of general relativity in 1915 can still be used for the local 
(non-cosmological) confirmed predictions; basically this is a matter of taking the low-energy Newtonian approximation we 
have derived from quantum gravity (based entirely on observables), converting it into tensor field form, and introducing 
field energy conservation. 

The Einstein-Hilbert gravitational field Lagrangian 

The Einstein-Hilbert action, S = J Ldt = J £d 4 x — }R(-g) ' r /(\6%G)d 4 x, when "varied" to find its minima, i.e. for the 
condition dS — (the Euler-Lagrange law), gives the basic field equation of general relativity). However, this free field 

Lagrangian £ = R(-g) ' r / (\6%G) (see for its derivation) is geometrically contrived to model 
accurately only the classical path of least action (i.e. the real or onshellpath in a path integral), unlike quantum field theory luigrangians which 
must be generally applicable for all paths. The way that general relativity is mathematically made into the Holy Grail of quantum 
gravity is obfuscating and misleading, spuriously focussing on a search for spin-2 graviton theories (which are merely con- 
sistent with the rank-2 field tensors of general relativity's field description), rather than on a search for theories which are 
consistent with the physical content of general relativity, i.e. the use of the metric in the relativistic correction to Newtonian 
gravitation which is required for conservation of energy. This relativistic correction produces the essential experimental 
checks on general relativity. Whether gravitation is described by curved spacetime using rank-2 tensors or with curved field 
lines using rank-1 tensors (vector calculus) is not empirically defensible physics. There is a confusion in the literature over 
which parts of general relativity are experimentally defensible; spin-2 gravitons are not experimentally defensible. We show 
how the metric can be used in with rank-1 tensors and spin-1 or even spin-0 gravitons to produce the same relativistic cor- 
rections to Newtonian gravity as are usually done using general relativity with rank-2 tensors and assumed spin-2 gravitons. 

Action is the Lagrangian energy density integrated over spacetime, which for a free field (with no matter) is given by grav- 
itational field energy density ( ): 

S = J £d 4 x = \R{-g) l / 2 c A /{\6KG)d A x, 

and the law of least action states that classical laws are recovered in the limit of least action, which must be an action min- 
ima where dS — (the Euler-Lagrange law): 

dS = J {d[R(-g) 1 / 2 c 4 ]/(16nGdg^ v )}dg^d 4 x= 0, 

hence the derivative d[R(-g) ' ]/dg^ = 0. Employing the product rule of differentiation gives: 

d[R{-g) l/2 Vdg^ = {-^^dR/dg^ + (-g) A / 2 Rd(-3 1/2 /4^- 

Therefore, in order that dS = 0, it follows that {-g)^ ' 2 dRl dg^ y + (-g)~~^> 2 Rd(-g)^' 2 /dJ JV = 0, where the partial derivative 
of the Ricci scalar is dR - Ruydgf^, and by Jacobi's formula dg = gg^ dgyy, so that (-g) ' Rd(-g) ' /dg^ = - iRguy 

4R(-i 1/2 ]/^ v = V - 2% v - 

This R„ v - zRgmj rigorously corrects R„ v = AnGTyy/tr. The celebrated 8%G/c multiplication factor of Einstein's field 
equation is not a G prediction, but is just the Newtonian law normali2ation for weak fields. Set R//v — ^&li\ = K Tuv an< ^ 
multiply out by^* (to give contractable tensor products): 

Introducing the scalars T = g^^uw an< ^ ^ = & ^UV an< ^ ^ e identities g^&uv = ^u = ^ ^ or 4-dimensional spacetime) 
and T = g Tqq = p, yields: 

R-4(sR) = KT 
r = _kT=-k/% =-K P . 


Putting this scalar curvature result into R/iv - ^&li\~ K ^UV an< ^ repeating the contraction procedure by multiplying out 

00 = 8 o c 

byg ® (note of course that ^Tnn = 8(P = 1) 

V = 2(" K PV + K V 


Rqq = 5(-Kp) + Kp = 5Kp. 

Thus, in the Newtonian (non-relativistic) limit, Rqq = 2Kp = V 2 k = 4nGp/c 2 , so 2^-9 = 4nGp/c 2 , or K = 8kG/c 2 . 

If a term for the kinetic energy of matter, _L is added to the free field Lagrangian for the action, the variation of the 
Lagrangian by amount d^ then produces a formula for the contributions by matter to the stress-energy tensor, Twy = - 
2(dL m /dgr- ) + g\x^ m Einstein's "cosmological constant," A (lambda), is included by changing the free field part of the 
Lagrangian toL= (R - 2A)<r ^(-g)^' 2 /(16llG), which yields: 

But A is not a checkable prediction in this equation, because it is not mechanistically linked to G, but instead is just an 
adjustable ad hoc parameter which reduces the checkable falsifiability of the theory. Einstein in his 1917 paper "Cosmological 
Considerations on the General Theory of Relativity" added A with a large positive (outward acceleration) value, to just can- 
cel out gravity at the average distance between galaxies, to keep the universe static (as then allegedly observed by 
astronomers). Beyond the average distance of separation of galaxies, repulsion predominated in Einstein's model. There 
are serious falsehoods in Einstein's A-based static universe. First, Alexander A. Friedmann in 1 922 showed it to be theo- 
retically unstable: any perturbation would cause the expansion or contraction of such Einstein's universe. 

Second, in 1929 Einstein's static universe was shown by Edwin Hubble's expansion evidence to be observationally false. 
Einstein then set A = 0, adopting the Friedmann-Robertson- Walker solution for the uniform curvature of a homogeneous, 

isotropic universe: k — R [87tGp/3) - H z ] where His Hubble's recession law parameter, H = v/R, and R = ct is the scale 
factor. In flat spacetime, k — 0, and the Einstein-de Sitter critical density (needed to just make the universe collapse, if grav- 

ity were a universal attractive force, rather than a mechanistic result of cosmological acceleration) is P cr itical = 3H /(87CG), so that the ratio 

of the actual mass density to the critical density in flat spacetime is D. = p/p cr iti C al = 87tGp/(3LP). The Friedmann- 
Lemaitre equation states: 

a = rh 2 = (R/3)(A - 87tGp) 

We define A as positive for outward acceleration. Readers will find other versions, where A is defined negative and multi- 
plied by ir to give energy density (not mass density), or where the geometric multiplier is 471 (for Newtonian non-relativis- 

tic motion) rather than 871 (for relativistic motion). 

Einstein's heuristic derivation of the basic field equation of general relativity 

Einstein, however, initially used an heuristic method to obtain and understand the basic field equation. In the Newtonian 
(or weak field) limit, gravitation is given by the scalar traces of the RJcci and stress-energy tensors (top-left to bottom-right diag- 
onal sums of the tensor matrices): 

R =/ fV V = R 00 + R // + R 22 + %i> 

For the non-relativistic Newtonian fall of an apple, Ricci's curvature is approximated by Poisson's law, Rqq ~ V k — 

4%Gp/c , while T — g Tqq = p. For radial symmetry about radius r, the Laplacian of k is V ^k — (a/r^ + (a/rS) + (a/r^) 

= 3a/ r. 

Einstein's field equation is derived by transforming Newton's law into a tensor spacetime curvature, with a correction for 
energy conservation. 

(1) Convert Newtonian gravity's Poisson law, V A k = AllGp/c , into a tensor equation by substituting \7 z k -* Rqq -» R„ v 

9 9 9 

and p -* Tqq -* T„ v , so that R„ v = AllGTyy/c (note that E — mc^ converts energy density p to mass density p/c ). 

(2) Recognise the local energy conservation error: both sides must have zero divergence, and while this is true for the Ricci 
tensor, V^R„ V = 0, it is not correct for the stress-energy tensor, V^T„ V ■£ 0. 

This makes R„ v = 4tcGT„ v /^ fail a self-consistency test, since both sides must have identical divergence, but they don't: 
V^Rwy ^ V"(47lGTiry/(T ). To give an example, the free electromagnetic field energy density component of the gravi- 
tational field source tensor is Tqq = (tE A + J3 z /|0,)/(87l), which generally has a divergence. 

(3) Correct R^y = 47lGT,,y/ c for local energy conservation by recognising that Bianchi's formula allows the replacement 
of the wrong divergence, V^T„ V ■£ 0, with: V^(T„ V — iTgipf) = 0, implying the stress-energy tensor correction, T„ v -* 
T /JV ~ 2 T ^V 

The term T^y — 2^ny ^ as zero divergence because subtracting 2^ny removes non-diverging components from the 

9 1 

stress-energy tensor, giving the correct formula, R/fv = (8kG/c )(Tmj ~ 2^™)i which is exactly equivalent to field equa- 
tion R^ y - 5R^ V = SnGT^/c 2 . 

Einstein originally used trial and error to discover this. In his 11 November 1915 communication to the Berlin Academy 
Einstein suggested that the solution is that the scalar trace, T, has zero divergence. But after correspondence with Hilbert 
who had ignored the physics and concentrated on the least action derivation, Einstein around 25 November 1915 realized 
from Bianchi's identity was compatible with Hilbert's tentative more abstract and guesswork mathematical approach, and 


the simplest correction is T„ v -* T„ v — iTguy, which gives zero divergence. In this nascent approach, Einstein was explor- 
ing various possibilities and trying out general ideas to solve problems, not working on an axiomatic proof. After Einstein had the insight 
from Bianchi's identity, he able to grasp the physical significance of the result from finding the least action to free-field "prop- 
er path" Lagrangian, Re (-g) ' /(167I&). This crucial background is often eliminated from textbooks on general relativity. 

The Einstein-Hilbert Lagrangian is that for relativistic Newtonian gravity, dealing with particles following the path of least action, on 
the relativistic mass shell, i.e. where curvature (field) always results from energy. By contrast, Tagrangians for fields in tested quantum 
field theories are based on non-classical Maxwell-Einstein and Yang-Mills field field potential amplitudes, which the Aharonov—Bohm effect jus- 
tifies, allow field energy to exist in space even where there is no directly measurable (onshell) electromagnetic field present, just energy due to can- 

celled-amplitude offshell field quanta. 

General relativity: energy always contributes to the gravitational field, hence all of it always flows along the path of least 
action. There is therefore deliberately no incorporation of other paths (hidden energy) in the Einstein-Hilbert Lagrangian 
of general relativity. This delusion is summed up in the field equation of general relativity, where all energy contributes to 
the field (curvature): no presence of energy is possible unless it contributes to the gravitational field. The whole definition 
of gravitational energy in general relativity limits the Lagrangian to describing only onshell energy and paths of least action. 
The Einstein-Hilbert Lagrangian is completely contrived and deluded, since it misses out all interference paths, which are 
all of the paths off the path of least action. 

Quantum field theory (path integral): energy doesn't always contribute to a field, because amplitudes far off the path of least action 
cancel one another, although offshell energy exchange still occurs along such paths, as shown by the Aharonov— Bohm effect. 
Although the original Maxwell-Heaviside equations were analogous to Newtonian gravity in so much as they merely mod- 
elled observable fields, Einstein in 1916 reformulated them into vector potential form, which includes hidden energy where 
amplitudes cancel, but particle paths (offshell energy) is still present. In understanding quantum force fields, the presence 
of the "cancelled" non-least action paths (ignored in general relativity) are vitally important. 

We in 1996 predicted the isotropic cosmological acceleration outwards a ~ He where His Hubble's empirical parameter 
(from his empirical law of galaxy cluster recession velocities, namely v — HR, where R is distance) from Hubble's law: 


v = HR = Ha past = Hc(H~ -t since big bang ), 

a = dp/dt since big bang = d [ He (H 4 - / since big bang ) ]/dt since big bang = - He = - 6.9 X 10" 10 ms" 2 . 

Because we have found that F stzv {ty = ma °/(4^R ), introduction of a — - He and rav ^, = K(2GM/r") from the obser- 
vations in Fig. 2 give: 

^gravity = ma °/(4tcR 2 ) = -mH (G 1 M 2 I(R 2 ? ). 

9 9 9 A 9 

Comparison of this result, J 7 y^, = -mHe (G M /(R c ), with the Newton-Laplace law Fgr^yit^ = M^MjG/R , shows 
that G = P / ' (mH) and M = M7M9, quantising mass into similar fundamental units ( ). 

This equation G = r /(mH) has been investigated by others independently since our publication in 1996. Louise Riofrio 

% ■ _i 

empirically formulated it as Gm — tc where / is the age of the universe (t — H ) without the quantum gravity theory, 

which shows that it is a valid and interesting result. She has suggested that the right hand side of Gm — tc is a constant, 

so that light velocity varies as inversely as the cube of time, c — (Gm/t)'. This is problemmatic theoretically and our 

approach is different, in that the time variation in Gm ~ tc is due to a direct proportionality between G and /. Gravitational 
interaction strength is predicted here to be increasing in direct proportion to the age of the universe, as demonstrated by 
evidence from the small fluctuations in the cosmic background radiation proving the flatness of the universe at early times. 

This flatness is due simply to a roughly 1,000 times weaker gravitational coupling at the 300,000 years CBR decoupling time, 
not to the theory of "inflation" which tried to reduce gravitational curvature by dispersing matter faster than the speed of 
light. Dirac investigated a time-varying G theory but assumed (wrongly) that it varied independently of the electromagnet- 
ic and strong couplings, which led Teller to point out in 1948 that the variation in forces in the sun would affect fusion rates 
incorrectly. All of the "checks" for time-variance of G are false because they make Teller's assumption (implicitly) that the 
gravitational coupling varies independently of the other force couplings, like electromagnetism. Force unification evidence 
suggests that all of the couplings vary in the same way. This negates the "no-go" G variation data of Teller and others. 

For example, the fusion rate in the sun depends on gravitational compression of protons overcoming their Coulomb repul- 
sion, if you merely vary G then fusion rates are affected, but if you double both gravity and Coulomb (both inverse square 
law forces), the relative increase in gravitational compression is offset by the relative increase in Coulomb repulsion. 

Implications for dogmatically believed classical approximations like general relativity 

Whenever a radically new idea that works comes along, people's first defense against progress is to point out that the new 
idea is "wrong" as judged by from the standpoint that the previous theory. In fact, disagreements between a new theory 
and an old theory are not detrimental if the new theory reproduces the empirically-confirmed predictions from the old theory in a new way. 
It is not true that every new theory must contain the old theory as a subset (e.g. thermodynamics doesn't include caloric). 
General relativity is today the gold standard in empirically validated gravitation, just as the Standard Model is the gold stan- 
dard in empirically validated electroweak and strong interactions. It is therefore important to set out precisely what parts 
of these theories, general relativity and the Standard Model of particle physics, are empirically defensible and how the new 
theory retains those empirically-defensible predictions from the older theories. 

General relativity survives as a classical approximation to gravity which makes Newtonian gravitation relativistic. All of the 
falsifiable predictions from general relativity such as light deflection, clock slowing in gravitational fields, excess radius, and 
gravitational redshift, stem from this relativistic correction, which exists in the mechanism of quantum gravity. General 
relativity does not correctly predict cosmological acceleration for the simple reason that it is based on Newton's delusion 
that gravity is a universal force, rather than being a Casimir type "attraction" due to repulsive effects originating in the dis- 
tant universe pushing together local masses (which non-receding relative to one another). This mechanism for quantum 
gravity invalidates the application of general relativity to cosmology. Although Einstein included an ad hoc "cosmological 
constant" in general relativity in 1917 to try to make it model the universe, this differs from the quantum gravity mecha- 
nism we offer, which links together dark energy and gravitation, a ~ c I ' (Gm) = 7 X 10 ms , or A = c '/(C <trr ). 

Feynman's "path integral" represents all of quantum mechanics complexity by interferences between amplitudes for small 
mechanical interactions, each represented by a Feynman diagram consisting of the exchange of a gauge boson which hits an 
effective interaction cross-section which is proportional to the square of the running coupling. The probability or relative 

cross-section for a reaction is proportional to | ¥ e ffective 1 2 = I ¥l + ¥2 + ¥3 + ¥4 + - 1 2 = \\ e ' Dx\ 2 , where \|/| , 
\|/ 2 , XI/3, and 11/4 are individual wavefunction path amplitudes representing all the different ways that gauge boson exchanges 

mediated forces between charges in the path integral, J e ' Dx. The complex wavefunction amplitude e where S is the 
path action in quantum action units, is unjustified by the successes of quantum field theory where measurables (probabil- 
ities or cross-sections) are real scalars. So the observable resultant arrow for a path integral on an Argand diagram must be 

always parallel to the real axis, thus instead of r as a unit length arrow with variable direction, can replace it by a single 

variable scalar quantity, e 1 ~* cos S, eliminating Hilbert space and Haag's theorem to renormali2ation. This reduction of 
quantum field theory to real space gives a provably self-consistent, experimentally checked quantum gravity. Path integral 

J e ' Dx is a double integral because action S is itself the integral of the lagrangian energy for a given Feynman dia- 
gram, which must be integrated over all paths not merely the classical path of least action, which only emerges classically 
as a result of multipath interferences, where paths with higher than minimal action cancel out. 

Lagrangian for quantum gravity and SU(2) Yang-Mills mechanism for electromagnetism 

Quantum gravity is a U(l) Abelian theory with only a single charge sign, which bypasses renormali2ation loop problems; 
there is no antigravity charge, preventing gravity-polari2ed pair production loops, so there is no running of the gravity coupling, 
thus quantum gravity renormali2ation is not required. Electromagnetism employs massless Yang-Mills SU(2) charged bosons 
(off-shell Hawking radiation). Cancellation of magnetic self-inductance for charged massless boson propagation necessi- 
tates a two-way exchange equilibrium of massless field quanta charge (the charge exchange equilibrium obviously doesn't 
extend to energy, since a particle's frequency can be redshifted to lower energy without any loss of electric charge), constrain- 
ing to 2ero the Yang-Mills net charge-tmmfei current, 2tAy X Fu V — 0, reducing the total Yang-Mills current Jj. + (2zAy X 

F„ v ) = -dF^y I ' dxy = -dyF^y to Maxwell's /„ = -dyF„y, so the Yang-Mills field strength F„ v = d^V a y - dyW a „ + 

&abc^' v^ u, l° ses lts term f° r the net transfer of charge, g£ a y c W^ 'yW c „ = 0, yielding Maxwell's F„ v = d^Ay — dyA„. 
Notice that the weak coupling, g, occurs in the disappearing charge transfer term. The mechanism eliminates the weak 
dependence on mass, turning a Yang-Mills theory into an effective Abelian one. 

Further details: